Consonants stay the same mostly, I haven't figured out what to do with /ʒ/, but it's somewhat marginal, and so I'm tempted to just leave /dʒ/ as it is.
Leave it as is, then. It derives from two sources: French loanwords, and palatalization of /zj/. Your spelling would thus reflect this.
I'm somewhat tempted to do something to mark the difference between voiced and unvoiced <th>, but I'm not sure. /k/ is a little problematic as well, my first instinct would be to use <k> for Germanic loans, then <c> for other loans, but Old English did use <c>...
You could use <k> everywhere except when doubled, in which case you'd use <ck>. This would follow the conventions used in German and Swedish.